Arithmetic groups with isomorphic finite quotients

Aka, Menny

doi:10.1016/j.jalgebra.2011.10.033

Cited by 24 publications

(70 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This was extended by Grunewald, Pickel and Segal to the class of all virtually polycyclic groups [GPS80]. Such a result also holds for finitely generated virtually free groups, and for some S-arithmetic groups [GZ11,Aka12a].…”

Section: A Rigidity Results With Respect To Dehn Fillingsmentioning

confidence: 77%

Recognizing a relatively hyperbolic group by its Dehn fillings

Dahmani¹,

Guirardel²

2018

Duke Math. J.

View full text Add to dashboard Cite

Dehn fillings for relatively hyperbolic groups generalize the topological Dehn surgery on a non-compact hyperbolic 3-manifold such as hyperbolic knot complements. We prove a rigidity result saying that if two non-elementary relatively hyperbolic groups without certain splittings have sufficiently many isomorphic Dehn fillings, then these groups are in fact isomorphic. Our main application is a solution to the isomorphism problem in the class of non-elementary relatively hyperbolic groups with residually finite parabolic groups and with no suitable splittings.

show abstract

Section: A Rigidity Results With Respect To Dehn Fillingsmentioning

confidence: 77%

Recognizing a relatively hyperbolic group by its Dehn fillings

Dahmani¹,

Guirardel²

2018

Duke Math. J.

View full text Add to dashboard Cite

show abstract

“…When the lattices also have the congruence subgroup property, one obtains nonisomorphic lattices with isomorphic profinite completions. These examples are not new as they appeared in [1]. The volume is not a purely local invariant and volumes of associated manifolds under our bijection do not always agree; see the example at the end of Section 3.…”

Section: Galois Cohomology Sets and Maximal Latticesmentioning

confidence: 83%

Locally Equivalent Correspondences

Linowitz¹,

McReynolds²,

Miller³

2017

Ann. inst. Fourier

View full text Add to dashboard Cite

Given a pair of number fields with isomorphic rings of adeles, we construct bijections between objects associated to the pair. For instance we construct an isomorphism of Brauer groups that commutes with restriction. We additionally construct bijections between central simple algebras, maximal orders, various Galois cohomology sets, and commensurability classes of arithmetic lattices in simple, inner algebraic groups. We show that under certain conditions, lattices corresponding to one another under our bijections have the same covolume and pro-congruence completion. We also make effective a finiteness result of Prasad and Rapinchuk.

show abstract

“…Our standing assumption in what follows is that there is exactly one 1 ≤ j ≤ d such that H ν j (R) ∼ = SU(n, 1) and that H ν k (R) ∼ = SU(n + 1) for all k = j. 1 The restriction of scalars…”

Section: Preliminariesmentioning

confidence: 99%

“…More recently, Bridson, McReynolds, Reid, and Spitler [7] showed that the arithmetic Kleinian group PSL 2 (Z[e 2πi/3 ]) is profinitely rigid. In the negative direction, Baumslag gave examples of nonisomorphic infinite finitely generated nilpotent groups with isomorphic profinite completions [2], and the congruence subgroup property leads to examples of nonisomorphic higher rank lattices in semisimple Lie groups with isomorphic profinite completions [1].…”

Section: Introductionmentioning

confidence: 99%

“…Results of this kind date back to work of Kazhdan in the early 70s [8], and we closely follow more recent work of Milne and Suh [11], who constructed examples of nonisomorphic higher-rank lattices with isomorphic profinite completions using conjugate but not homeomorphic Shimura varieties. (In contrast with [1], they do not need to assume the congruence subgroup property holds.) The main contribution of this paper is to show that their use of the Mar-gulis superrigidity theorem can be weakened to Mostow rigidity, which then applies for PU(n, 1).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation